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On Thermalization in Classical Scalar Field Theory

Gert Aarts A, Gian Franco Bonini, CHRISTOF WETTERICH

Published Jan 1, 2000• 34 citations
OpenAlex

Abstract

Thermalization of classical fields is investigated in a \\phi^4 scalar field theory in 1+1 dimensions, discretized on a lattice. We numerically integrate the classical equations of motion using initial conditions sampled from various nonequilibrium probability distributions. Time-dependent expectation values of observables constructed from the canonical momentum are compared with thermal ones. It is found that a closed system, evolving from one initial condition, thermalizes to high precision in the thermodynamic limit, in a time-averaged sense. For ensembles consisting of many members with the same energy, we find that expectation values become stationary - and equal to the thermal values - in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In contrast, fluctuations and connected correlation functions will differ substantially from the canonical values. This raises doubts on the use of a straightforward expansion in terms of 1PI-vertex functions to study thermalization.

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The approach to thermalization in the classical phi^4 theory in 1+1 dimensions: energy cascades and universal scaling

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We study the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 1+1 spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. We find that the Hartree approximation is remarkably accurate in equilibrium. The non-equilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities.The time evolution is described by several stages with a cascade of energy towards the ultraviolet. After a transient stage, the spatio-temporal gradient terms become larger than the nonlinear term and a stage of universal cascade emerges.This cascade starts at a time scale t_0 independent of the initial conditions (except for very low energy density). Here the power spectra feature universal scaling behavior and the front of the cascade k(t) grows as a power law k(t) sim t^alpha with alpha lesssim 0.25. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperatureTeff(t) which slowly decreases as sim t^{-alpha}.Two well separated time scales emerge while Teff(t) varies slowly, the wavectors in the wake with k < k(t) attain LTE on much shorter time scales.This universal scaling stage ends when the front of the cascade reaches the cutoff at a time t_1 sim a^{-1/alpha}. Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.

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Temperature and conditions for thermalization after canonical quenches

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Remarks on thermalization in 2D CFT

Jan De Boer A, Dalit Engelhardt

We revisit certain aspects of thermalization in 2D conformal field theory (CFT). In particular, we consider similarities and differences between the time dependence of correlation functions in various states in rational and non-rational CFTs. We also consider the distinction between global and local thermalization and explain how states obtained by acting with a diffeomorphism on the ground state can appear locally thermal, and we review why the time-dependent expectation value of the energy-momentum tensor is generally a poor diagnostic of global thermalization. Since all 2D CFTs have an infinite set of commuting conserved charges, generic initial states might be expected to give rise to a generalized Gibbs ensemble rather than a pure thermal ensemble at late times. We construct the holographic dual of the generalized Gibbs ensemble and show that, to leading order, it is still described by a Banados-Teitelboim-Zanelli black hole. The extra conserved charges, while rendering $c<1$ theories essentially integrable, therefore seem to have little effect on large-$c$ conformal field theories.

arXiv:1604.05327•Physical review. D/Physical review. D.•Dec 30, 2016•41 citations
Thermalization of classical fields is investigated in a \phi^4 scalar field theory in 1+1 dimensions, discretized on a lattice. We numerically integrate the classical equations of motion using initial conditions sampled from various nonequilibrium probability distributions. Time-dependent expectation values of observables constructed from the canonical momentum are compared with thermal ones. It is found that a closed system, evolving from one initial condition, thermalizes to high precision in the thermodynamic limit, in a time-averaged sense. For ensembles consisting of many members with the same energy, we find that expectation values become stationary - and equal to the thermal values - in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In contrast, fluctuations and connected correlation functions will differ substantially from the canonical values. This raises doubts on the use of a straightforward expansion in terms of 1PI-vertex functions to study thermalization.
We study the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 1+1 spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. We find that the Hartree approximation is remarkably accurate in equilibrium. The non-equilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities.The time evolution is described by several stages with a cascade of energy towards the ultraviolet. After a transient stage, the spatio-temporal gradient terms become larger than the nonlinear term and a stage of universal cascade emerges.This cascade starts at a time scale t_0 independent of the initial conditions (except for very low energy density). Here the power spectra feature universal scaling behavior and the front of the cascade k(t) grows as a power law k(t) sim t^alpha with alpha lesssim 0.25. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperatureTeff(t) which slowly decreases as sim t^{-alpha}.Two well separated time scales emerge while Teff(t) varies slowly, the wavectors in the wake with k < k(t) attain LTE on much shorter time scales.This universal scaling stage ends when the front of the cascade reaches the cutoff at a time t_1 sim a^{-1/alpha}. Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.
We study the dynamics of thermalization and the approach to equilibrium in the classical ${\ensuremath{\varphi}}^{4}$ theory in 1+11+11+1 spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. In this context, comparing to the exact results we find that the Hartree approximation is remarkably accurate in equilibrium. The nonequilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities. The long time evolution is described by several distinct stages, all characterized by a cascade of energy towards the ultraviolet. After an initial transient stage, the spatiotemporal gradient terms become larger than the nonlinear term, and there emerges a stage of universal cascade. This cascade starts at a time scale t0{t}_{0}t0​ independent of the initial conditions (except for very low energy density). During this stage the power spectra feature universal scaling behavior and the front of the cascade k‾(t)\overline{k}(t)k(t) moves to the ultraviolet as a power law $\overline{k}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\lesssim}0.25$ an exponent weakly dependent on the energy density alone. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperature Teff(t),{T}_{\mathrm{eff}}(t),Teff​(t), which slowly decreases with time as $\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\alpha}}.$ Two well separated time scales emerge: while Teff(t){T}_{\mathrm{eff}}(t)Teff​(t) varies slowly, the wave vectors in the wake with k<k‾(t)k<\overline{k}(t)k<k(t) attain LTE on much shorter time scales. This universal scaling stage ends when the front of the ultraviolet cascade reaches the cutoff at a time scale ${t}_{1}\ensuremath{\sim}{a}^{\ensuremath{-}1/\ensuremath{\alpha}}.$ Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.
We describe a Hartree ensemble method to approximately solve the Heisenberg equations for the ϕ 4 model in 1 + 1 dimensions. We compute the energies and number densities of the quantum particles described by the ϕ field and find that the particles initially thermalize with a Bose-Einstein distribution for the particle density. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved. 1. Hartree ensemble approximation Non-perturbative equilibrium properties of quantum field theories are studied with great success using numerical lattice techniques. Similarly one would like to investigate quantum field dynamics non-perturbatively. This has obvious applications to e.g. heavy ion collisions and early universe physics. Furthermore, it could be used to study equilibrium properties in cases where Monte Carlo methods do not work, such as in QCD at finite density or chiral gauge theories. Using Monte Carlo techniques to compute the Minkowski time path integral is, of course, not possible. Neither can one directly solve the Heisenberg equations for the quantum field operators. Hence one has to resort to approximations, such as classical dynamics [2] (for recent work see [3]), large n or Hartree (see e.g. [4]). Here we discuss a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. Ref. [1] also contains a discussion of this method and a detailed presentation will appear elsewhere. We use the lattice ϕ4 model in 1+1 dimensions as a test model, which has the following Heisenberg operator equations, ˙ˆϕ = ˆπ, ˙ ˆπ = ( ∆ − µ
When a nonintegrable system evolves out of equilibrium for a long time, local observables are in general expected to attain stationary expectation values, independent of the details of the initial state. But the thermalization of a closed quantum system is not yet well understood. Here we show that it presents indeed a much richer phenomenology than its classical counterpart. Using a new numerical technique, we identify two distinct regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well known in classical systems, shows convergence to thermal values only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.
We consider quenches of a quantum system that is prepared in a canonical equilibrium state of one Hamiltonian and then evolves unitarily in time under a different Hamiltonian. Technically, our main result is a systematic expansion of the pre- and postquench canonical ensembles in the quench strength. We first demonstrate how this can be used to predict the system's temperature after the quench from equilibrium properties at the prequench temperature. For a thermalizing postquench system, it furthermore allows us to calculate equilibrium observable expectation values. Finally, in the presence of additional conserved quantities besides the Hamiltonian, we obtain a hierarchy of necessary conditions for thermalization towards the (postquench) canonical ensemble. At first order, these thermalization conditions have a nice geometric interpretation in operator space with the canonical covariance as a semi-inner product: The quench operator (difference between post- and prequench Hamiltonians) and the conserved quantity must be orthogonal in the orthogonal complement of the postquench Hamiltonian. We illustrate the results numerically for a variety of setups involving integrable and nonintegrable models.
We revisit certain aspects of thermalization in 2D conformal field theory (CFT). In particular, we consider similarities and differences between the time dependence of correlation functions in various states in rational and non-rational CFTs. We also consider the distinction between global and local thermalization and explain how states obtained by acting with a diffeomorphism on the ground state can appear locally thermal, and we review why the time-dependent expectation value of the energy-momentum tensor is generally a poor diagnostic of global thermalization. Since all 2D CFTs have an infinite set of commuting conserved charges, generic initial states might be expected to give rise to a generalized Gibbs ensemble rather than a pure thermal ensemble at late times. We construct the holographic dual of the generalized Gibbs ensemble and show that, to leading order, it is still described by a Banados-Teitelboim-Zanelli black hole. The extra conserved charges, while rendering c<1c<1c<1 theories essentially integrable, therefore seem to have little effect on large-ccc conformal field theories.